Sorting is a very classic problem of reordering items (that can be compared, e.g. integers, floating-point numbers, strings, etc) of an array (or a list) in a certain order (increasing, non-decreasing, decreasing, non-increasing, lexicographical, etc).

There are many different sorting algorithms, each has its own advantages and limitations.

Sorting is commonly used as the introductory problem in various Computer Science classes to showcase a range of algorithmic ideas.

Without loss of generality, we assume that we will sort only Integers, not necessarily distinct, in non-decreasing order in this visualization. Try clicking Bubble Sort for a sample animation of sorting the list of 5 jumbled integers (with duplicate) above.

Click 'Next' (on the top right)/press 'Page Down' to advance this e-Lecture slide, use the drop down list/press 'Space' to jump to a specific slide, or Click 'X' (on the bottom right)/press 'Esc' to go to Exploration mode.

Pro-tip: Since you are not logged-in, you may be a first time visitor who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown] to advance to the next slide, [PageUp] to go back to the previous slide, [Esc] to toggle between this e-Lecture mode and exploration mode.

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When an (integer) array A is sorted, many problems involving A become easy (or easier):

Searching for a specific value v in array A,

Finding the min/max or the k-th smallest/largest value in (static) array A,

Testing for uniqueness and deleting duplicates in array A,

Counting how many time a specific value v appear in array A,

Set intersection/union between array A and another sorted array B,

Finding a target pair x ∈ A and y ∈ A such that x+y equals to a target z, etc.

Discussion: In real-life classes, the instructor may elaborate more on these applications.

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Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2017). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this.

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There are two actions that you can do in this visualization.

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The first action is about defining your own input, an array/a list that is:

In Exploration mode, you can experiment with various sorting algorithms provided in this visualization to figure out their best and worst case inputs.

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The second action is the most important one: Execute the active sorting algorithm by clicking "Sort" menu and then clicking "Go".

Remember that you can switch active algorithm by clicking the respective abbreviation on the top side of this visualization page.

Some sorting algorithms have certain additional options. You may toggle the options as you wish before clicking "Go". For example, in Bubble Sort (and Merge Sort), there is an option to also compute the inversion index of the input array (this is an advanced topic).

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View the visualisation/animation of the chosen sorting algorithm here.

Without loss of generality, we only show Integers in this visualization and our objective is to sort them from the initial state into ascending order state.

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At the top, you will see the list of commonly taught sorting algorithms in Computer Science classes. To activate each algorithm, select the abbreviation of respective algorithm name before clicking "Sort → Go".

To facilitate more diversity, we randomize the active algorithm upon each page load.

The first six algorithms are comparison-based sorting algorithms while the last two are not. We will discuss this idea midway through this e-Lecture.

The middle three algorithms are recursive sorting algorithms while the rest are usually implemented iteratively.

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To save screen space, we abbreviate algorithm names into three characters each:

Comparison-based Sorting Algorithms:

BUB - Bubble Sort,

SEL - Selection Sort,

INS - Insertion Sort,

MER - Merge Sort (recursive implementation),

QUI - Quick Sort (recursive implementation),

R-Q - Random Quick Sort (recursive implementation).

Not Comparison-based Sorting Algorithms:

COU - Counting Sort,

RAD - Radix Sort.

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We will discuss three comparison-based sorting algorithms in the next few slides:

You should see a 'bubble-like' animation if you imagine the larger items 'bubble up' (actually 'float to the right side of the array').

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Comparison and swap require time that is bounded by a constant, let's call it c.

There are two nested loops in (the standard) Bubble Sort.

The outer loop runs for exactly N iterations.But the inner loop runs get shorter and shorter:

When i=0, (N−1) iterations (of comparisons and possibly swaps),

When i=1, (N−2) iterations,...,

When i=(N−2), 1 iteration,

When i=(N−1), 0 iteration.

Thus, the total number of iterations = (N−1)+(N−2)+...+1+0 = N*(N−1)/2 (derivation).

Total time = c*N*(N−1)/2 = O(N^2).

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Bubble Sort is actually inefficient with its O(N^2) time complexity. Imagine that we have N = 105 numbers. Even if our computer is super fast and can compute 108 operations in 1 second, Bubble Sort will need about 100 seconds to complete.

However, it can be terminated early, e.g. try Bubble Sort on the small sorted ascending example shown above [3, 6, 11, 25, 39] where it terminates in O(N) time.

The improvement idea is simple: If we go through the inner loop with no swapping at all, it means that the array is already sorted and we can stop Bubble Sort at that point.

Discussion: Although it makes Bubble Sort runs faster in general cases, this improvement idea does not change O(N^2) time complexity of Bubble Sort... Why?

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Given an array of N items and L = 0, Selection Sort will:

Find the position X of the smallest item in the range of [L...N−1],

Swap X-th item with the L-th item,

Increase the lower-bound L by 1 and repeat Step 1 until L = N-2.

Without further ado, let's try Selection Sort on the same small example array [29, 10, 14, 37, 13].

Without loss of generality, we can also implement Selection Sort in reverse:Find the position of the largest item Y and swap it with the last item.

These sorting algorithms are usually implemented recursively, use Divide and Conquer problem solving paradigm, and run in O(N log N) time for Merge Sort and O(N log N) time in expectation for Randomized Quick Sort.

PS: The the non-randomized version of Quick Sort runs in O(N2) though.

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Given an array of N items, Merge Sort will:

Merge each pair of individual element (which is by default, sorted) into sorted arrays of 2 elements,

Merge each pair of sorted arrays of 2 elements into sorted arrays of 4 elements,Repeat the process...,

Final step: Merge 2 sorted arrays of N/2 elements (for simplicity of this discussion, we assume that N is even) to obtain a fully sorted array of N elements.

This is just the general idea and we need a few more details before we can discuss the true form of Merge Sort.

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We will dissect this Merge Sort algorithm by first discussing its most important sub-routine: The O(N) merge.

Given two sorted array, A and B, of size N1 and N2, we can efficiently merge them into one larger combined sorted array of size N = N1+N2, in O(N) time.

This is achieved by simply comparing the front of the two arrays and take the smaller of the two at all times. However, this simple but fast O(N) merge sub-routine will need additional array to do this merging correctly. See the next slide.

Contrary to what many other CS printed textbooks usually show (as textbooks are static), the actual execution of Merge Sort does not split to two subarrays level by level, but it will recursively sort the left subarray first before dealing with the right subarray.

In Merge Sort, the bulk of work is done in the conquer/merge step as the divide step does not really do anything (treated as O(1)).

When we call merge(a, low, mid, high), we process k = (high-low+1) items.There will be at most k-1 comparisons.There are k moves from original array Is There Shopper Mile Statement Extra Never Go The Bag Crowded Natural Tote a to temporary array b and another k moves back.In total, number of operations inside merge sub-routine is < 3k-1 = O(k).

The important question is how many times this merge sub-routine is called?

There are log N levels and in each level, we perform O(N) work, thus the overall time complexity is O(N log N). We will later see that this is an optimal (comparison-based) sorting algorithm, i.e. we cannot do better than this.

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The most important good part of Merge Sort is its O(N log N) performance guarantee, regardless of the original ordering of the input. That's it, there is no adversary test case that can make Merge Sort runs longer than O(N log N) for any array of N elements.

Merge Sort is therefore very suitable to sort extremely large number of inputs as O(N log N) grows much slower than the O(N2) sorting algorithms that we have discussed earlier.

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Quick Sort is another Divide and Conquer sorting algorithm (the other one discussed in this visualization page is Merge Sort).

We will see that this deterministic, non randomized version of Quick Sort can have bad time complexity of O(N2) on adversary input before continuing with the randomized and usable version later.

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Divide step: Choose an item p (known as the pivot)Then partition the items of a[i..j] into three parts: a[i..m-1], a[m], and a[m+1..j].a[i..m-1] (possibly empty) contains items that are smaller than p.a[m] is the pivot p, i.e. index m is the correct position for p in the sorted order of array a.a[m+1..j] (possibly empty) contains items that are greater than or equal top.Then, recursively sort the two parts.

Conquer step: Don't be surprised... We do nothing :O!

If you compare this with Merge Sort, you will see that Quick Sort D&C steps are totally opposite with Merge Sort.

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We will dissect this Quick Sort algorithm by first discussing its most important sub-routine: The O(N) partition (classic version).

After this, a[2] = 27 is guaranteed to be sorted and now Quick Sort recursively sorts the left side a[0..1] first and later recursively sorts the right side a[3..5].

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First, we analyze the cost of one call of partition.

Inside partition(a, i, j), there is only a single for-loop that iterates through (j-i) times. As j can be as big as N-1 and i can be as low as 0, then the time complexity of partition is O(N).

Similar to Merge Sort analysis, the time complexity of Quick Sort is then dependent on the number of times partition(a, i, j) is called.

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When the array a is already in ascending order, like the example above, Quick Sort will set p = a[0] = 5, and will return m = 0, thereby making S1 region empty and S2 region: Everything else other than the pivot (N-1 items).

Try Quick Sort on example input array [5, 18, 23, 39, 44, 50].

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On such worst case input scenario, this is what happens:

The first partition takes O(N) time, splits a into 0, 1, N-1 items, then recurse right.The second one takes O(N-1) time, splits a into 0, 1, N-2 items, then recurse right again....Until the last, N-th partition splits a into 0, 1, 1 item, and Quick Sort recursion stops.

This is the classic N+(N-1)+(N-2)+...+1 pattern, which is O(N2), similar analysis as the one in this slide...

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The best case scenario of Quick Sort occurs when partition always splits the array into two equal halves, like Merge Sort.

When that happens, the depth of recursion is only O(log N).

As each level takes O(N) comparisons, the time complexity is O(N log N).

Same as Quick Sort except just before executing the partition algorithm, it randomly select the pivot between a[i..j] instead of always choosing a[i] (or any other fixed index between [i..j]) deterministically.

Try Random Quick Sort on this large and somewhat random example array.

Mini exercise: Implement the idea above to the implementation shown in this slide!

It will take about 1 hour lecture to properly explain why this randomized version of Quick Sort has expected time complexity of O(N log N) on any input array of N elements.

In this e-Lecture, we will assume that it is true.

If you need non formal explanation: Just imagine that on randomized version of Quick Sort that randomizes the pivot selection, we will not always get extremely bad split of 0 (empty), 1 (pivot), and N-1 other items. This combination of lucky (half-pivot-half), somewhat lucky, somewhat unlucky, and extremely unlucky (empty, pivot, the rest) yields an average time complexity of O(N log N).

Discussion: Actually the phrase "any input array" above is not fully true. There is actually a way to make the randomized version of Quick Sort as currently presented in this VisuAlgo page still runs in O(N2). How?

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Outer Material: Cotton

Material Composition: 100% Cotton

We will discuss two non comparison-based sorting algorithms in the next few slides:

These sorting algorithms can be faster than the lower bound of comparison-based sorting algorithm of Ω(N log N) by not comparing the items of the array.

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It is known (also not proven in this visualization as it will take another 1 hour lecture to do so) that all comparison-based sorting algorithms have a lower bound time complexity of Ω(N log N).

Thus, any comparison-based sorting algorithm with worst-case complexity O(N log N), like Merge Sort is considered an optimal algorithm, i.e. we cannot do better than that.

However, we can achieve faster sorting algorithm — i.e. in O(N) — if certain assumptions of the input array exist and thus we can avoid comparing the items to determine the sorted order.

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Assumption: If the items to be sorted are Integers with small range, we can count the frequency of occurrence of each Integer (in that small range) and then loop through that small range to output the items in sorted order.

Try Counting Sort on the example array above where all Integers are within [1..9], thus we just need to count how many times Integer 1 appears, Integer 2 appears, ..., Integer 9 appears, and then loop through 1 to 9 to print out x copies of Integer y if frequency[y] = x.

The time complexity is O(N) to count the frequencies and O(N+k) to print out the output in sorted order where k is the range of the input Integers, which is 9-1+1 = 9 in this example. The time complexity of Counting Sort is thus O(N+k), which is O(N) if k is small.

We will not be able to do the counting part of Counting Sort when k is relatively big due to memory limitation, as we need to store frequencies of those k integers.

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Assumption: If the items to be sorted are Integers with large range but of few digits, we can combine Counting Sort idea with Radix Sort to achieve the linear time complexity.

In Radix Sort, we treat each item to be sorted as a string of w digits (we pad Integers that have less than w digits with leading zeroes if necessary).

For the least significant (rightmost) digit to the most significant digit (leftmost), we pass through the N items and put them according to the active digit into 10 Queues (one for each digit [0..9]), which is like a modified Counting Sort as this one preserves stability. Then we re-concatenate the groups again for subsequent iteration.

Try Radix Sort on the example array above for clearer explanation.

Notice that we only perform O(w × (N+k)) iterations. In this example, w = 4 and k = 10.

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There are a few other properties that can be used to differentiate sorting algorithms on top of whether they are comparison or non-comparison, recursive or iterative.

In this section, we will talk about in-place versus not in-place, stable versus not stable, and caching performance of sorting algorithms.

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A sorting algorithm is said to be an in-place sorting algorithm if it requires only a constant amount (i.e. O(1)) of extra space during the sorting process. That's it, a few, constant number of extra variables is OK but we are not allowed to have variables that has variable length depending on the input size N.

Merge Sort (the classic version), due to its merge sub-routine that requires additional temporary array of size N, is not in-place.

A sorting algorithm is called stable if the relative order of elements with the same key value is preserved by the algorithm after sorting is performed.

Example application of stable sort: Assume that we have student names that have been sorted in alphabetical order. Now, if this list is sorted again by tutorial group number (recall that one tutorial group usually has many students), a stable sort algorithm would ensure that all students in the same tutorial group still appear in alphabetical order of their names.

Discussion: Which of the sorting algorithms discussed in this e-Lecture are stable?Try sorting array A = {3, 4a, 2, 4b, 1}, i.e. there are two copies of 4 (4a first, then 4b).

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e-Lecture: The content of this slide is hidden and only available for selected CS lecturer who knows Dr Steven Halim personally.

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Actually, the C++ source code for many of these basic sorting algorithms are already scattered throughout these e-Lecture slides. For other programming languages, you can translate the given C++ source code to the other programming language.

Usually, sorting is just a small part in problem solving process and nowadays, most of programming languages have their own sorting functions so we don't really have to re-code them unless absolutely necessary.

This is not the end of the topic of sorting. When you explore other topics in VisuAlgo, you will realise that sorting is a pre-processing step for many other advanced algorithms for harder problems, e.g. as the pre-processing step for Kruskal's algorithm, creatively used in Suffix Array data structure, etc.

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As the action is being carried out, each step will be described in the status panel.

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Control the animation with the player controls! Keyboard shortcuts are:

Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

About

✕

VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace.

VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim) and beyond. Today, some of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Though specifically designed for National University of Singapore (NUS) students taking various data structure and algorithm classes (e.g. CS1010, CS1020, CS2010, CS2020, CS3230, and CS3230), as advocators of online learning, we hope that curious minds around the world will find these visualisations useful too.

VisuAlgo is not designed to work well on small touch screens (e.g. smartphones) from the outset due to the need to cater for many complex algorithm visualizations that require lots of pixels and click-and-drag gestures for interaction. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly.

VisuAlgo is an ongoing project and more complex visualisations are still being developed.

The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms. The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. This online quiz system, when it is adopted by more CS instructors worldwide, should technically eliminate manual basic data structure and algorithm questions from typical Computer Science examinations in many Universities. By setting a small (but non-zero) weightage on passing the online quiz, a CS instructor can (significantly) increase his/her students mastery on these basic questions as the students have virtually infinite number of training questions that can be verified instantly before they take the online quiz. The training mode currently contains questions for 12 visualization modules. We will soon add the remaining 8 visualization modules so that every visualization module in VisuAlgo have online quiz component.

List of translators who have contributed ≥100 translations can be found at statistics page.

Acknowledgements This project is made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).

Terms of use

✕

VisuAlgo is free of charge for Computer Science community on earth. If you like VisuAlgo, the only payment that we ask of you is for you to tell the existence of VisuAlgo to other Computer Science students/instructors that you know =) via Facebook, Twitter, course webpage, blog review, email, etc.

If you are a data structure and algorithm student/instructor, you are allowed to use this website directly for your classes. If you take screen shots (videos) from this website, you can use the screen shots (videos) elsewhere as long as you cite the URL of this website (http://visualgo.net) and/or list of publications below as reference. However, you are NOT allowed to download VisuAlgo (client-side) files and host it on your own website as it is plagiarism. As of now, we do NOT allow other people to fork this project and create variants of VisuAlgo. Using the offline copy of (client-side) VisuAlgo for your personal usage is fine.

Note that VisuAlgo's online quiz component is by nature has heavy server-side component and there is no easy way to save the server-side scripts and databases locally. Currently, the general public can only use the 'training mode' to access these online quiz system. Currently the 'test mode' is a more controlled environment for using these randomly generated questions and automatic verification for a real examination in NUS. Other interested CS instructor should contact Steven if you want to try such 'test mode'.

List of Publications

This work has been presented briefly at the CLI Workshop at the ACM ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012).

VisuAlgo is not a finished project. Dr Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Dr Steven Halim. His contact is the concatenation of his name and add gmail dot com.